Properties

Label 4730.2.a.r
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} - q^{5} + \beta q^{6} + ( -2 - \beta ) q^{7} + q^{8} + ( 2 + \beta ) q^{9} +O(q^{10})\) \( q + q^{2} + \beta q^{3} + q^{4} - q^{5} + \beta q^{6} + ( -2 - \beta ) q^{7} + q^{8} + ( 2 + \beta ) q^{9} - q^{10} - q^{11} + \beta q^{12} -4 q^{13} + ( -2 - \beta ) q^{14} -\beta q^{15} + q^{16} + ( 1 - \beta ) q^{17} + ( 2 + \beta ) q^{18} + ( 3 - \beta ) q^{19} - q^{20} + ( -5 - 3 \beta ) q^{21} - q^{22} + 4 q^{23} + \beta q^{24} + q^{25} -4 q^{26} + 5 q^{27} + ( -2 - \beta ) q^{28} + ( -2 - 2 \beta ) q^{29} -\beta q^{30} -4 q^{31} + q^{32} -\beta q^{33} + ( 1 - \beta ) q^{34} + ( 2 + \beta ) q^{35} + ( 2 + \beta ) q^{36} -4 q^{37} + ( 3 - \beta ) q^{38} -4 \beta q^{39} - q^{40} + ( 2 - 2 \beta ) q^{41} + ( -5 - 3 \beta ) q^{42} + q^{43} - q^{44} + ( -2 - \beta ) q^{45} + 4 q^{46} + ( -2 + \beta ) q^{47} + \beta q^{48} + ( 2 + 5 \beta ) q^{49} + q^{50} -5 q^{51} -4 q^{52} + ( 1 - 3 \beta ) q^{53} + 5 q^{54} + q^{55} + ( -2 - \beta ) q^{56} + ( -5 + 2 \beta ) q^{57} + ( -2 - 2 \beta ) q^{58} + ( -8 + 3 \beta ) q^{59} -\beta q^{60} + ( 6 - 2 \beta ) q^{61} -4 q^{62} + ( -9 - 5 \beta ) q^{63} + q^{64} + 4 q^{65} -\beta q^{66} -8 q^{67} + ( 1 - \beta ) q^{68} + 4 \beta q^{69} + ( 2 + \beta ) q^{70} + ( -14 + \beta ) q^{71} + ( 2 + \beta ) q^{72} + ( -2 - 2 \beta ) q^{73} -4 q^{74} + \beta q^{75} + ( 3 - \beta ) q^{76} + ( 2 + \beta ) q^{77} -4 \beta q^{78} + ( -6 - \beta ) q^{79} - q^{80} + ( -6 + 2 \beta ) q^{81} + ( 2 - 2 \beta ) q^{82} + ( 4 + \beta ) q^{83} + ( -5 - 3 \beta ) q^{84} + ( -1 + \beta ) q^{85} + q^{86} + ( -10 - 4 \beta ) q^{87} - q^{88} + ( -6 + 6 \beta ) q^{89} + ( -2 - \beta ) q^{90} + ( 8 + 4 \beta ) q^{91} + 4 q^{92} -4 \beta q^{93} + ( -2 + \beta ) q^{94} + ( -3 + \beta ) q^{95} + \beta q^{96} + ( -2 + 4 \beta ) q^{97} + ( 2 + 5 \beta ) q^{98} + ( -2 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + q^{3} + 2q^{4} - 2q^{5} + q^{6} - 5q^{7} + 2q^{8} + 5q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + q^{3} + 2q^{4} - 2q^{5} + q^{6} - 5q^{7} + 2q^{8} + 5q^{9} - 2q^{10} - 2q^{11} + q^{12} - 8q^{13} - 5q^{14} - q^{15} + 2q^{16} + q^{17} + 5q^{18} + 5q^{19} - 2q^{20} - 13q^{21} - 2q^{22} + 8q^{23} + q^{24} + 2q^{25} - 8q^{26} + 10q^{27} - 5q^{28} - 6q^{29} - q^{30} - 8q^{31} + 2q^{32} - q^{33} + q^{34} + 5q^{35} + 5q^{36} - 8q^{37} + 5q^{38} - 4q^{39} - 2q^{40} + 2q^{41} - 13q^{42} + 2q^{43} - 2q^{44} - 5q^{45} + 8q^{46} - 3q^{47} + q^{48} + 9q^{49} + 2q^{50} - 10q^{51} - 8q^{52} - q^{53} + 10q^{54} + 2q^{55} - 5q^{56} - 8q^{57} - 6q^{58} - 13q^{59} - q^{60} + 10q^{61} - 8q^{62} - 23q^{63} + 2q^{64} + 8q^{65} - q^{66} - 16q^{67} + q^{68} + 4q^{69} + 5q^{70} - 27q^{71} + 5q^{72} - 6q^{73} - 8q^{74} + q^{75} + 5q^{76} + 5q^{77} - 4q^{78} - 13q^{79} - 2q^{80} - 10q^{81} + 2q^{82} + 9q^{83} - 13q^{84} - q^{85} + 2q^{86} - 24q^{87} - 2q^{88} - 6q^{89} - 5q^{90} + 20q^{91} + 8q^{92} - 4q^{93} - 3q^{94} - 5q^{95} + q^{96} + 9q^{98} - 5q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.79129
2.79129
1.00000 −1.79129 1.00000 −1.00000 −1.79129 −0.208712 1.00000 0.208712 −1.00000
1.2 1.00000 2.79129 1.00000 −1.00000 2.79129 −4.79129 1.00000 4.79129 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4730.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.r 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)
\(43\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\( T_{3}^{2} - T_{3} - 5 \)
\( T_{7}^{2} + 5 T_{7} + 1 \)
\( T_{13} + 4 \)